To find the inverse of the function $f(x) = \sqrt[3]{1 - x}$, we need to solve for $x$ in terms of $y$ and then interchange $x$ and $y$.

Step-by-Step Solution

1. Start with the function: $y = \sqrt[3]{1 - x}$

2. Isolate the cube root: $y^3 = 1 - x$

3. Solve for $x$: $x = 1 - y^3$

4. Interchange $x$ and $y$ to find the inverse function: $y = 1 - x^3$

5. Write the inverse function: $f^{-1}(x) = 1 - x^3$

Final Answer

The inverse function is: $f^{-1}(x) = 1 - x^3$

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Related Questions

1. How do you verify if two functions are inverses of each other?
2. What are the conditions for a function to have an inverse?
3. How do you find the domain and range of an inverse function?
4. What is the graphical relationship between a function and its inverse?
5. How does the process of finding an inverse change for different types of functions (e.g., quadratic, exponential)?

Tip

To find an inverse function, remember to switch $x$ and $y$ after solving for $y$ in terms of $x$. This helps to represent the inverse function correctly.